RAIROMODÉLISATION MATHÉMATIQUE

ET ANALYSE NUMÉRIQUE

I. GASSER

R. ILLNER

P. A. MARKOWICH

C. SCHMEISERSemiclassical, t → ∞ asymptotics and dispersive effectsfor Hartree-Fock systemsRAIRO — Modélisation mathématique et analyse numérique, tome 32, no 6 (1998), p. 699-713.<http://www.numdam.org/item?id=M2AN_1998__32_6_699_0>

© SMAI, EDP Sciences, 1998, tous droits réservés.

L’accès aux archives de la revue « RAIRO — Modélisation mathématiqueet analyse numérique » (http://www.esaim-m2an.org/) implique l’accord avecles conditions générales d’utilisation (http://www.numdam.org/legal.php). Touteutilisation commerciale ou impression systématique est constitutive d’une in-fraction pénale. Toute copie ou impression de ce fichier doit contenir laprésente mention de copyright.

Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiques

http://www.numdam.org/

FT MATHEMATICAL MODELLING AND NUHERICAL ANALYSIS[MODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE

(Vol 32, n° 6, 1998, p 699 à 713)

SEMICLASSICAL, f - *~ ASYMPTOTICS AND DISPERSIVE EFFECTS FOR HARTREE-FOCK SYSTEMSDedicated to Helmut Neunzert at the occasion of his 60th birthday

I. GASSER (*), R. ILLNER (**), P. A. MARKOWICH (*) and C. SCHMEISER (***)

Résumé — On analyse la limite semiclassique et Vasymptotique t —> °° pour des systèmes des équations de Schrodinger faiblement nonlinéaire en forme Hartree-Fock En utilisant des techniques de fonction de Wigner, on démontre que la limite semi-classique est représentéepar l'équation de Vlasov « s elf-consistent » En outre, on démontre des estimations du temps pour la densité et le potentiel électrique deHartree-Fock dans les norms if pour t —> » © Elsevier, Pans

Abstract — We analyze the semiclassical limit and the "t —> °° asymptotics" ofmildly nonhnear Schrodinger Systems of (self-consistent)Hartree-Fock form Using Wigner-functwn techniques we prove that the semiclassical limit is represented by the self-consistent Vlasovéquation Moreover we prove time decay for the position density and for the Hartree-potential in Lp norms as t —> » © Elsevier, Pans

1. INTRODUCTION

We consider Hartree-Fock Systems in Ud of the form

2ie -r- we, = — -=- -

2 X' V\(x, f) y/), x e W, t e U, l e N (Lia). 7 = 1

, leN (1.1b)

=| U(x-z)n(z,t)dz (l.ld)d

^ , 0 = 1 U(x-z)y/el(z,t)ye(z,t)dz. (1.1e)

Hère e > 0 dénotes the scaled Planck-constant, X]^ 0 the occupation number of the state y/ev n is the number

density of the considered particle System, V8H is the self-consistent Hartree potential (defined by the interaction

potential U = U(x)), V^ represents a given exterior potential and Vetj stands for the interaction of the l-th and

j-th state.

Manuscnpt received April 1, 1997(H) FB Mathematik MA 6-2, TU-Berlm, Stra(3e des 17 Juni 136, 10623 Berlin, Germany(xx) Department of Mathematics and Statistics, University of Victoria, RO Box 3045, Victoria, B.C. V8W 3P4, Canada.(***) Institut fur Angewandte und Numensche Mathematik, TU-Wien5 Wiedner Hauptstrape 8-10, 1040 Wien Austria

M2 AN Modélisation mathématique et Analyse numérique 0764-583X/98/06Mathematical Modelling and Numencal Analysis © Elsevier, Pans

700 I. GASSER, R. ILLNER, P. A. MARKOWICH, C. SCHMEISER

Hartree-Fock Systems are considered an accurate description of the quantum-mechanical évolution of a Fermionsystem, since their dérivation from many body phy sics takes into account the Pauli exclusion principle [ S ] , whichis not the case for Hartree Systems (obtained by setting V^:=0).

In this paper we consider two limits of Hartree-Fock Systems. The first one, analyzed in Section 2, is thesemiclassical limit s —» 0. We prove — under suitable assumptions on the data — that the Hartree-Fock exchangeterm does not give a contribution in the limit £ —» 0, i.e. the semiclassical limit of the Hartree-Fock system is —in a sense made précise in the next section — the selfconsistent Vlasov équation. The same result has alreadybeen shown for Hartree Systems [LPa, MM]. Clearly, this behaviour is physically plausible, since the Pauliprinciple is a purely quantum physical notion.

The second limit to be considered is the limit t —> oo in the purely répulsive case U ̂ 0. These results, whichimprove [DF] are contained in Section 3.

Section 4 is concerned with dispersive effects.

2. THE SEMICLASSICAL LIMIT

We define the density matrix p£ in the usual way

/(r,M) = i^;(r ,0^(^) , r9seUd (2.1)

and dérive the Heisenberg formulation of the Hartree-Fock system:

(VeH(r,t)-Vls

H(s,t))pe

(U(r-z)-U(s-z))p\r,z>t)pXz,s,t)dz (2.2a)

n\x,t)=p\x,x,t) (2.2b)

-I.

40,0= U(x-z)n(z,t)dz (2.2c)Jud

z

p£(r,s, t=0) = ̂ X](p](r) (p](s) = \ p](r> s) . (2.2d)

The Wigner transform of the density matrix is the Fourier transform of the function

pEl x + 2J?'x~2ï1*t) w ^ r e s P e c t t o */> ^-e-

we(x, v, t) := ^1 pel x + 2 tf* x~~ ô 7» f ) eWV n drj (2.3)

{cf. [GMMP], [LPa], [W], where the Fourier transform is defined by

)ew xdx. (2.4){Infini

M2 AN Modélisation mathématique et Analyse numériqueMathematical Modelling and Numencal Analysis

SEMICLASSICAL, t -> <*> ASYMPTOTICS AND DISPERSIVE EFFECTS FOR HARTREE-FOCK SYSTEMS 701

It is the solution of the Wigner-Hartree-Fock équation, obtained from (2.2) by an easy calculation [M]:

xwe + e\VE1 wE + 6\Ve

H] we + £2e[_we] = 0 ,

VH(x,t)=\ U(x-z)n\z,t)dz,

Jut

n\x,t)=\ w\x,v,t)dv,

(2.5a)

(2.5b)

(2.5c)

(2.5d)

For a given potential V = V( x ) the pseudo-differential operator ö£ [ V] is defined by

where >v dénotes the inverse Fourier transform of w-w(x, v) with respect to v:

W(JC, ^ ) = vw(x ïv)é~ iV '7 dv .Ju-

r3e is the (quadratically) nonlinear operator

The following estimate is basic for carrying out the limit e —» 0 + in the Hartree-Fock system.

LEMMA 2.1: Let we L2(Udxxüj), ç>s S(Ud

x X

f Ö6C dv

and

A\cp)

on Ud. Then

wffA y/( ri ) := sup | <p( x, n ) |.

vol. 32, n" 6, 1998

(2.6a)

(2.6b)

(2.7)

(2.8a)

(2.8b)

702 I. CASSER, R. ILLNER, P. A. MARKOWICH, C. SCHMEISER

Proof: With the substitution s = -z + %> r = z + % w e obtain

Qe[w] (j>dxdv = i\ [ed~l(U(sr) - U(es))] w(jt + | J , r ) W(JC - | r, s) <p(x9 r + s) dr ds dx

We estimate

tpdxdv

( |w(jc, drds

X M \w(x,s)\2dx\ (j\wUr)\2dx) drds

and thus

dx dv « 2 l\U(er)\fdr\\w{x,.1/2

f |M>(X, \r)\2dxdrds

The assertion of the Lemma now follows immediately. DThe subséquent Lemma is conceraed with a priori conserved quantities of the Hartree-Fock system:

LEMMA 2.2: Let U(x) = U(-x) on Ud hold. Then

n(x,t)dx=\ nI(x)dx, Vf G M ( charge conservation) , (2.9)

where nj(x) :=2

Ee(t)=E£(0), VfeR, (2.10)

M2 AN Modélisation mathématique et Analyse numériqueMathematical Modelling and Numerical Analysis

SEMICLASSICAL, t -» « ASYMPTOTICS AND DISPERSIVE EFFECTS FOR HARTREE-FOCK SYSTEMS 703

where

U(x-z)n(xft)nXz,t)dxdz

U(x-z) \pË(x,z,t)\2dxâz

(energy conservation).

Proof: (2.9) is obtained by multiplying the Hartree-Fock équation (1.1a) by X]y/], integrating by parts, takingimaginary parts and summing over /.

(2.10) is the result of a somewhat more tedious calculation based on multiplying (1.1a) by X] -jr y/p taking realparts and summing over L Details can be found in [CG] (at least for the Coulomb interaction potentialU(x)=T^ronU3). D

EE(t) is the total energy, which by effect of Lemma 2.2 is constant in time. We remark that, by a well knowncalculation (see, e.g. [MM], [LPa]) the kinetic energy can be written as

z Jndï=i| f Q (2.11)

The folio wing Lemma pro vides an a priori L2 -estimate for the Wigner function ws :

LEMMA 2.3: Assume that the initial states {ç?/}"=1 are an orthonormal system in L (U.x) and thatU(x) = U(-x) on Md. Then

Proof: Similarly to the proof of (2.9) we show that initially orthogonal states (pev (p\ remain orthogonal for all

time under the Hartree-Fock évolution. The result then follows directly from the formulas (2.3), (2.1). DAlso, we remark that the local conservation law

^ e = 0 (2.13)

holds, where the current density f can be calculated from the Wigner function in the usual way

Je(x, t) = vw£(x, v, t) dv (2.14)

(see [MM, LPa]). We now make the following assumptions on the data:(Al) (i) Ve e (0, e0], l e N : X]> 0 ;

(ii) Vee (0, e0] : {ç^i i s a n O N S i n ̂ 2 /

(iii) 3 O 0 :

vol. 32, n° 6, 1998

704 I. GASSER, R. ILLNER, R A. MARKOWICH, C. SCHMEISER

forée (0,e0].On the external potential we assume

(A2) VEe Hll0C(Ud) ; 3Ve M : VE(x) ^ V°n Ud,

and on the interaction potential:(A3) (i) U(x) = U(-x) on Ud

(ii)

' 2<5<oo if d = 29

Ue Lr(Rd) + Ls'"(Rd) with < | ^ s < ~ > i f d = 3 ,

C/e C&(R) and U(0) = 0 if rf= 1 ;

(iii) Vf/ G L̂ rfT8~( Ud) + L*°°( Ud) wi th 2 / ++

88 < q < 2.

For the définition of the 'weak Lp -spaces' Lp' we refer to [RS, page 30]. (Al) implies a uniform bound forne LT(nt;L\n*)) for the initial kinetic energy and (with (A3) (i)) on w3 e L°°(Ut ; L

2(Udx Ud)).In order to carry out the Hmit s —> 0 in the Wigner-Hartree-Fock System we proceed as in [LPa] to establish

uniform a priori bounds. We start with the initial energy:

PROPOSITION 2.1: EE(0) ^ C.

From now on we dénote by C generic, not necessarily equal constants which are independent ofs e (0 f e 0 ] .

Proof: The following estimate can be found in [LPa] (cf. the Theorem in the Appendix)

(2.15)

where Co is also independent of w£ and

0= 4

d + 4'

d + 4 ,

Evaluating at t = 0 gives a uniform bound for nr G Ld + 2(Ux). The generalized Young inequah'ty [RS, page 32]

then yields (together with (A3) (ii)) a uniform bound for |C/ (x-z) | n}(x) n)(z) dzdx. Since (by theSchwartz inequality)

ne(x,t)n\z9t)& \pe(x9z9t)\2 (2.16)

obtain a uniform bound for | U(x- z)| \p)(x,z)\2 dzdx and the assertion of Proposition 2.1 follows,J ni x M*

wex

Next we dérive a uniform bound for the total kinetic energy:

PROPOSITION 2.2:

=\ teUt.

M2 AN Modélisation mathématique et Analyse numénqueMathematical Modelling and Numerical Analysis

SEMICLASSICAL, t -> « ASYMPTOTICS AND DISPERSIVE EFFECTS FOR HARTREE-FOCK SYSTEMS 705

Proof: From (2.10), (2.11) we obtain

E^t) £(f = O ) - V n(t)dx

- i f U+(x-z) (n\x, t) n\z, t) - \p\x, z, t)\2) dz dx

+ \ f t IT(x-z) (n(x, t) n\z, t) - \p\x, z, t)\2) dzdx

and (2.15) gives

EL0) ^ c + \ \ „ d \U{x-z)\ n{x,t)n\z,t)dzdx.

For d = 1 the assertion follows. For d > 1 we again apply the generalized Young inequality

f d\U(x-z)\n(x,t)n(z,t)dzdx^ C( 1 + \\n(0]\2LP,

1 2with 2 = - + - where s is of (A3) (ii). By interpolation we have

and (2.15) gives

and 2( 1 - 0, ) ( 1 - 0) = / ï " < 1 by (A3) (ü). Doo d + 4 d

We thus obtain a uniform bound for n G L°°( M.t ; Ld+2( Mx) ) from (2.15) and uniform bounds for

fand

inL"(R,) follow.Finally we need

f \U(x-z)\ \pe(x,z,t)\2dzdx

LEMMA 2.4: Let (A3) (i), (Ui) hold and assume that we G LT{ Ut ; L

2( M? x IR )̂ ) uniformly as e —> 0.

vol. 32, n° 6, 1998

706 I CASSER, R ILLNER, P A. MARKOWICH, C SCHMEISER

Proof: The assertion for d=\ follows immediately from Lemma2.1. Thus, we assume d>\ for thefollowing.

At first we observe that the L°°( Ud) part of U(x) gives an O(ed~1 ) contribution to A£(<P ) defined in (2.8) (b)and consequently by (2.8) (a) its contribution to £2Ê[v/] in S" is of the same order. Therefore, to complete theproof, it suffices to assume U e Z/'°°([Rrf) with s as of (A3) (ii).

We dénote Zs(x) := \ed"1 U(ex)\2 and estimate the convolution in (2.8b) using the generahzed Younginequality:

for y/ e Ll(Ud ) r\L°°(Ud), where 1 < q < <*>, 0<S<- and pà-—T~,—^r—r. Keeping q fixed and taking

ô to zero gives

II v * z 8 n j . . .

Since

we conclude the assertion with s = 2q, DThe existence of a unique solution of the Hartree-Fock problem (or, equivalently, the Wigner-Hartree-Fock

System) for e > 0 can easily be shown by generalizing the methods of [CG]. Details are left to the reader. Thelimit e —> 0 can now be carried out by applying the methods that lead to Theorem IV.5 in [LPa].

THEOREM 2.1: Let (Al), (A2), (A3) hold. Then, for every séquence e —> 0 there exists a subsequence (denotedby the same symbol) such that

w)-> w°j^ 0 in L2( Udx x Ud

v ) weakly , (2.17a)

w£ -> w° ^ 0 in L~( Rt ; L2( Ud X Ud

v ) ) weak- * , (2.17b)

ne ->n°=\w° dv in L°°( Ut ; Ld7i( Rd ) ) weak- * , (2.17c)

f _> 7° = f Ï;W° di? |W L°°( Kr ; L ^ ( Ud ) ) wcofe- * , (2.17d)

V Z C/(JC-Z)«°(Z,O& w L°°(nt;L2(nd))weak-*, (2.17e)

( w , n , £"° = VV^) are weak solutions of the self consistent Vlasov équation:

w°t+v- Vxw°-VxV°H • Vvw° = Q inUdxUdxUt (2.18a)

w°(t = 0)=w°I. (2.18b)

M2 AN Modélisation mathématique et Analyse numériqueMathematical Modelhng and Numencal Analysis

SEMICLASSICAL, t -» <~ ASYMPTOTICS AND DISPERSIVE EFFECTS FOR HARTREE-FOCK SYSTEMS 707

Remark 2.1: The important case of the Coulomb interaction in 3 dimensions

C/( J C)=-i-9 xG U3 (2.19)

is contained in the assumptions. We then have q = j and s = 3 in (A3). The limiting problem (2.18), (2.17c),(2.17e) is the Vlasov-Poisson équation [LPe], [LPa].

Remark 2.2: The case of the Poisson interaction U(x) = |x| in 1 dimension is not included because of (A3)(iii), which was imposed in order to be able to treat the (relatively uninteresting) 1-dimensional case analogouslyto the case d > 1. However, the assumption U e Cb( IR ) can easily be replaced by at most polynomial growthat oo and continuity at 0.

Remark 23: Both the attractive case ( U ^ 0) and the répulsive case ( U 2= 0) are covered by Theorem 2.1.

Remark 2.4: The cases of e-independent occupation probabilities X\ and of finitely many states (i.e. X] = 0 for/ > N) is not included because of (Al) (iii). While it can be dealt with rather easily in the Hartree case with asmooth interaction potential U (cf. [LPa]), it créâtes difficulties for the Hartree-Fock problem when the completesemiclassical information is sought. Then the Schrödinger problem (1.1) has to be dealt with as a fully coupledsystem of TV équations and methods as presented in [GMMP] have to be applied (passage to the semiclassical limitin the Wignermatrix of the Schrödinger system). Serious mathematical difficulties then occur at points in(x, ?)-space where the spectral décomposition of the Hartree interaction potential matrix V]. dégénérâtes. To ourknowledge, this problem has not been solved yet.

However, the semiclassical limit of ws(t) (and consequently of ne(t) and Je(t)) can still be computed.Therefore, assume that t / e C (IR) for some 0 < / ? ^ 1. A simple modification of the proof of Lemma 2.1shows that (2.8a) also holds with

Thus, by the regularity of U and since V£/(0) = 0 we obtain

Ae(<p) = O(ed+fi) inS(MdxUd). (2.20)

Instead of the uniform bound on — 2 (X\)2 in (iii) assume now that 2 i^])2 is bounded uniformly in £ (e.g.8 i = i i = i , v

finitely many states only). Lemma 2.3 then implies ||w£(t) \\L\udxUd) = ol — ) and Lemma 2.1 (with (2.20)

instead of (2.8b)) gives

The other terms in (2.5a) and (2.5c) can be taken to the limit as in [LP]. Thus, Theorem 2.1 also applies for smoothinteraction potential (instead of (A3)) without thprocess (2.17) have to be changed accordingly.interaction potential (instead of (A3)) without the uniform L2 -bound on ws, however the topologies for the limit

3. ASYMPTOTIC BEHAVÏOUR AS t -> <*> IN THE REPULSIVE CASE

In this section we investigate the time decay properties of the Hartree-Fock-System (2.5). We shall assumevanishing external potential VE = 0.

vol. 32, n° 6, 1998

708 I. GASSER, R. ILLNER, P. A. MARKOWICH, C. SCHMEISER

Also, we assume that a global unique strong solution of the Hartree-Fock-System exists. The assumptions ofthe previous section are sufficient for this; we remark that the (Al) (iii) can be weakened. In addition we imposethe following assumptions on the interaction potential:(A3) (iv) U=U0(\x\)&Q.

(v) ^ ( r ) ^ - ^ £ / 0 ( r ) , r > 0 , a > 0 .Also, for the sake of clarity óf the présentation we consider the case d ^ 2.

Note that results along the lines of the ones presented below entirely based on the Schrödinger formalismrestricted to the 3d Coulomb case and finitely many coupled states can be found in [DF, P]. Decay results for theHartree case with Coulomb interaction can be found in [ELZ].

We state

LEMMA 3.1: The following relation holds:

( l -hf 2 ~ a ) , a < 2

with c independent of t.

Proof: Using the équation (2.5a) we obtain

vt\2w\t)dxdv4 | f \x-dtiudjud

= - f f \x-vt\2{Ge[V£H] we

= 2t\ f x-v{eë[VeH]we + Q

~t2\ f \v\2{&s[VBH]w£

e] \dxdv

}dxdv.

An easy but tedious calculation gives

2t\ f x-v{€>£[VeH]we + Qe[we] }dxdv =

JRÎJMÎ

tl f z<VU(z){nE(x-z)nE(x)-\p£(x-z,x)\2}dzdx.Jud

xJudv

Now, combining the energy conservation (2.10) with VE = 0 and (2.11) gives

it\\\ \ \v\2™edxdv+\\ f U(x-z){ne(x)nXz)~\pe(x,z)\2}dxdz]=0.

With the relation

hél f v2wdxdv+\\ f \v\2{e£[V£H]

zatJudJui zJudJui(3.2)

M2 AN Modélisation mathématique et Analyse numériqueMathematical Modelling and Numerical Analysis

SEMICLASSICAL, t -» <*> ASYMPTOTICS AND DISPERSIVE EFFECTS FOR HARTREE-FOCK SYSTEMS 709

obtained by multiplying the équation (2.5a) by ^ |i;|2 and integrating over IRf x R j , we conclude

-t2] f \v\2{0e[VeH]w£ + Q£[we] }dxdv

= -t2i\ f U(x-z){nXx)n\z)-\pe(x9z)\2}dxdz.ai 0UdJ Ud

Therefore

Adt

• vt\2 w8 dx dv + g(t)f i , -

f (x-z)VU(x-z){nE(x)nE(z)-\p\x,z)\2}dxdz +ud

xJmdv l

= t2ï f U(x-z){nXx)ne(z)-\p\x,z)\2}dxdz

with

holds. Using the assumptions on the interaction potential we have

x- VxU+aU=r- U\r) + odJ ̂ 0.

We obtain

1 (2-Adt f f \x-

JudJuivt\2w£dxdv

o,

It is (again an easy but tedious calculation)

O I I ^ R - ) = f f \x- vt\2w£dxdv.

Therefore we can apply Gronwall's lemma to (3.3) and arrive at (3.1).We need also

LEMMA 3.2: = 0 and 2 ^ll-xp* llL2(Rf) < o ° - ^ estimate

holds for the wavefunctions corresponding to the solution of problem (2.5).

Proof: It is

IJCI n( jc,

(3.3)

D

vol. 32, n° 6, 1998

710 I. GASSER, R. HXNER, P. A. MARKOWICH, C. SCHMEISER

and

-™ \x\2n(x,t)dx = 2 X'Je(x,t)dxdtJudx Jud

=5 2 \x\2n\x,t)dx}

VL^ 2 VEE(O) - / I \x\2 ne(x, t) dx .

Then the result follows applying the Gronwall's lemma. D

This result is needed in order to apply the well known.

LEMMA 3.3: Let ue Hl(Ud) such that xu e L2(Ud). Then

\\ue\\LP^C(P,d) \\G(-t)ue\\l* Kx+ieNx)ue\\^af-X8a-1

where G(t) is the unitary group generated by the homogeneous linear Schrödinger équation,

2 =g p sg ̂ 2 and a is

The proof of this lemma can be found in [ILZ] or [GV].At this point we can state our decay result.

THEOREM 3.1: Under the assumptions of this section the following decay estimâtes hold:

(i) II n

c(te) 2U a\ a<2

I c(te)~ZKi~a) a ^ 2

(iü) 11^11^)= j (t x-a(i-fl)' ^ 9I c(t£) , a < 2

wïï / i 1—a = « ( l — — ) , — = — + ^-, 1 + — = — + ^r aw6? c independent of s. It is2 \ q / s q 2 r q d

1 ^ q ^ -3 ^, 1 ^ 51 < ̂ —5 T and

V

Proof: Following [ILZ] and using the Lemmas 3.1-3.3 we estimate

\-a

t m = 1

M2 AN Modélisation mathématique et Analyse numériqueMathematica! Modelling and Numencal Analysis

SEMICLASSICAL, t -> °° ASYMPTOTICS AND DISPERSIVE EFFECTS FOR HARTREE-FOCK SYSTEMS

and the decay result of the density follows. The decay resuit for the current is obtained using

711

^ n^ V^IIL^, * s K

Vil "'(O II *.(,*) ,

since e|| V^ | | L 2 ( R ^ is uniformly bounded by the energy conservation (2.10) (with VE = 0). The estimate (iii)follows using the Sobolev inequality

Judv

(y)h(y)dy f f "'WJuilui \x- y\

with - + 1 + ̂ = 2. Therefore,/ q d

and the assertion follows. D

4. A DISPERSIVE IDENTITY

Let xo€ Rd fixed with rf>l, setv(x-x0)

• a 1/a gives the identity:

= 0 or ô~\ and a > 0. Then multiplying (2.5a) by

ir, iuiiuw(x,v, dvdxdt

^ f f (x-xo)VU(x-z) 7

i—-^ K—^-{\p\x,Ztt)\2~n\x,t)n\z,t))dzdxdt

(x-x0)—^•(Js(x,T1)-J

£(x,T2))dx (4.1)

for all - oo < Tx < T2 < °o. Intégral identities of this type were obtained in [LPel] for the free transport équationand in [P] for the Vlasov-Poisson and Wigner-Poisson Systems.

vol. 32, n° 6, 1998

712 I- GASSER, R. ILLNER, P. A. MARKOWICH, C. SCHMEISER

A lengthy calculation shows that the first term on the left hand side of (4.1) is nonnegative. For example inthe case d = 3 and ô = 0 it is equal to

'zi^r[j.(g^f- i(^^T')i>+8-'^')'^' <«>i=i JT,\_JU3\ \X x o i |JC — JCO| /

(see [LPel] also for the other cases). Assume now that VE = 0 (no exterior field) and that the interaction potentialis radial U=UQ(\x\) with U'o(r)^O. Then, an easy calculation using pB{x, z, t) = pe(z, x, t) and (2.16)shows that also the third term in (4.1) is nonnegative. Thus, the identity (4.1) gives the bound for the first termon its left hand side:

UA^ii^j+iirc^iifcRï)-

Energy conservation shows that \\Je(t) \\L\ui) *s uniformly bounded in e and t. Thus, we conclude for d = 3 andail x0 G U3 :

^g^nj t f(j z^- i ( i"^Lv^*o i>*+ ' 'r.^' )*< c- c ( ' '- j t o» <4-3>

(just as for the free Schrôdinger équation). Similar estimâtes can be obtained for dimensions différent from 3.Other applications of the dispersive identity (4.1) are also possible.

ACKNOWLEDGEMENTS

This research was supported by grant Nr. A7487 of the Natural Science and Engineering Research Council ofCanada. The first and the third author were supported by grant Nr. MA 1662/1-1 and 1-2 of the german 'DeutscheForschungsgemeinschafV. The last author was supported by grant Nr. P11308-MAT of the austrian 'Found zurFôrderung der Wissenschaft und Forschung'. We would like to thank Dr. Raimund Wegener who alerted us to theproblems posed by the Hartree Fock Systems. I. Gasser, P. A. Markowich and C. Schmeiser would also like toacknowledge the hospitality of the University of Victoria, where this research was started.

REFERENCES

[CG] J. M. CHADAM and R. T. GLASSEY, Global existence of solutions to the Cauchy problem for time dépendent

Hartree équations, J.M.P. 16(5), pp. 1122-1130, 1975.

[DF] J. P. DlAS and M. FlGUEIRA, Conservation laws and time decayfor the solutions ofsome nonlinear Schrôdinger-

Hartree équations and Systems, J. Math. Anal. Appl. 84, pp. 486-508, 1981.

[GMMP] P. GERARD, P. A. MARKOWICH, N. J. MAUSER and F. POUPAUD, Homogenization limits and Wigner transforms,

Comm. Pure Appl. Math. 50, No. 4, pp. 323-379, 1997.

[GV] J. GlNtBRE and G. VELO, On a class of nonlinear Schrôdinger équations. IL Scattering theory, gênerai case, J.

Funct Anal. 32, pp. 33-71, 1979.

[ILZ] R. ILLNER, P. F. ZWEEFEL and H. LANGE, Uniqueness and asymptotic behaviour of solutions of the Wigner-

Poisson and the Schrôdinger-Poisson Systems, M2AS 17, pp. 349-376, 1994.

[LPa] P. L. LIONS et T. PAUL, Sur les mesures de Wigner, Revista Matematica Iberoamericana 9, pp. 553-618, 1993.

[LPe] P. L. LIONS et B. PERTHAME, Global solutions of Vlasov-Poisson type équations, preprint n. 8824, Ceremade,

1995.

[LPel] P. L. LIONS et B. PERTHAME, Lemme de moments, de moyenne et de dispersion, C. R. Acad. Sci. Paris, 314,Série 1, pp. 801-806, 1992.

M2 AN Modélisation mathématique et Analyse numériqueMathematical Modelling and Numerical Analysis

SEMICLASSICAL, t -> °° ASYMPTOTICS AND DISPERSIVE EFFECTS FOR HARTREE-FOCK SYSTEMS 713

[M] R A. MARKOWICH, On the équivalence of the Schrödinger and the Quantum Liouville équation, M2AS 11,

pp. 459-469, 1989.

[MM] P. A. MARKOWICH and N. J. MAUSER, Tlie classical limit of a self-consistent quantum-Vlmov équation in 3d,

M3AS 3, pp. 109-124, 1993.

[P] B. PERTHAME, Time decay, propagation of low moments and dispersive effects of kinetic équations, Com. PDE

21, No. 3-4, pp. 659-686, 1996.

[RS] M. REED and B. SIMON, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness,

Academie Press, New York, 1975.

[S] J. C. SLATER, A simplification of the Hartree-Fock method, Phys. Rev. 81(3), pp. 385-390, 1951.

[W] E. WiGNER, On the Quantum Correction for the Thermodynamic Equilibrium, Phys. Rev. 40, pp. 749-759, 1932.

vol. 32, n° 6, 1998